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Tight Algorithm for Connected Odd Cycle Transversal Parameterized by Clique-width

Narek Bojikian, Stefan Kratsch

TL;DR

This work provides a single-solution Monte Carlo algorithm for Connected Odd Cycle Transversal parameterized by clique-width that runs in $O^*(12^{\operatorname{cw}})$. It extends the Cut&Count framework via action-sequences and a parity-based representation scheme over a lattice of connectivity patterns, using fast lattice-convolution to achieve the claimed time bound. The authors prove SETH-tight lower bounds ruling out faster $O^*((12-\varepsilon)^{\operatorname{lcw}})$ algorithms and thereby close the second open question of Hegerfeld and Kratsch for this problem. The results advance the understanding of tight bounds for connectivity problems parameterized by clique-width and outline a path for applying similar techniques to other problems such as Feedback Vertex Set.

Abstract

Recently, Bojikian and Kratsch [2023] have presented a novel approach to tackle connectivity problems parameterized by clique-width ($\operatorname{cw}$), based on counting small representations of partial solutions (modulo two). Using this technique, they were able to get a tight bound for the Steiner Tree problem, answering an open question posed by Hegerfeld and Kratsch [ESA, 2023]. We use the same technique to solve the Connected Odd Cycle Transversal problem in time $\mathcal{O}^*(12^{\operatorname{cw}})$. We define a new representation of partial solutions by separating the connectivity requirement from the 2-colorability requirement of this problem. Moreover, we prove that our result is tight by providing SETH-based lower bound excluding algorithms with running time $\mathcal{O}^*((12-ε)^{\operatorname{lcw}})$ even when parameterized by linear clique-width. This answers the second question posed by Hegerfeld and Kratsch in the same paper.

Tight Algorithm for Connected Odd Cycle Transversal Parameterized by Clique-width

TL;DR

This work provides a single-solution Monte Carlo algorithm for Connected Odd Cycle Transversal parameterized by clique-width that runs in . It extends the Cut&Count framework via action-sequences and a parity-based representation scheme over a lattice of connectivity patterns, using fast lattice-convolution to achieve the claimed time bound. The authors prove SETH-tight lower bounds ruling out faster algorithms and thereby close the second open question of Hegerfeld and Kratsch for this problem. The results advance the understanding of tight bounds for connectivity problems parameterized by clique-width and outline a path for applying similar techniques to other problems such as Feedback Vertex Set.

Abstract

Recently, Bojikian and Kratsch [2023] have presented a novel approach to tackle connectivity problems parameterized by clique-width (), based on counting small representations of partial solutions (modulo two). Using this technique, they were able to get a tight bound for the Steiner Tree problem, answering an open question posed by Hegerfeld and Kratsch [ESA, 2023]. We use the same technique to solve the Connected Odd Cycle Transversal problem in time . We define a new representation of partial solutions by separating the connectivity requirement from the 2-colorability requirement of this problem. Moreover, we prove that our result is tight by providing SETH-based lower bound excluding algorithms with running time even when parameterized by linear clique-width. This answers the second question posed by Hegerfeld and Kratsch in the same paper.
Paper Structure (18 sections, 33 theorems, 66 equations, 1 figure)

This paper contains 18 sections, 33 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs, clique expressions and odd cycle transversals.
  4. Algebraic notation, sets, functions and lattices.
  5. Connectivity patterns
  6. The Algorithm
  7. Indices of the algorithm
  8. Description of the Algorithm
  9. Solution representation
  10. Action-sequences
  11. Solution representation through action-sequences.
  12. The Algorithm counts action-sequences
  13. Isolation lemma
  14. Correctness of the algorithm
  15. Lower bound
  16. ...and 3 more sections

Key Result

Theorem 1.1

There exists a one-sided error Monte-Carlo algorithm (only false negatives), that given a graph $G$ together with a $k$-expression of $G$, and a positive integer $\overline{b}$, solves the Connected Odd Cycle Transversal problem in time $\mathcal{O}^*(12^k)$, and outputs the correct answer with prob

Figures (1)

  • Figure 1: On the left, the graph $G$ corresponding to an instance with two clauses. The gray squares represent path gadgets. Decoding gadgets are depicted in cyan, and clause gadgets as circles. We outline in yellow, blue, green, purple, orange and brown, the first path sequence, bundle, path bundle, column, segment and section respectively. On the right we depict a simple path gadget. Subdivision vertices are drawn as circles, while triangle vertices are drawn as small gray disks.

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Folklore
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6
  • proof
  • ...and 69 more